How deep we can see

So here is a new approach. For a the same matrix of periods and wave heights, now I scanned at which depth the error Delta h becomes larger than the bar height, which I set to 1 m.



The result is shown in the image above and it can also be seen than longer waves with small nonlinearity are more likely to detect the bar at deeper absolute depths.
It is interesting that the depths are relatively small though, which may indicate that the technique is more suitable in relatively shallow waters. This can be seen if the same critical depth is used to calculate kh, as shown below.